On operators with the same local spectra

Torgašev, Aleksandar. "On operators with the same local spectra." Czechoslovak Mathematical Journal 48.1 (1998): 77-83. <http://eudml.org/doc/30403>.

@article{Torgašev1998,
abstract = {Let $B(X)$ be the algebra of all bounded linear operators in a complex Banach space $X$. We consider operators $T_1,T_2\in B(X)$ satisfying the relation $\sigma _\{T_1\}(x) = \sigma _\{T_2\}(x)$ for any vector $x\in X$, where $\sigma _T(x)$ denotes the local spectrum of $T\in B(X)$ at the point $x\in X$. We say then that $T_1$ and $T_2$ have the same local spectra. We prove that then, under some conditions, $T_1 - T_2$ is a quasinilpotent operator, that is $\Vert (T_1 - T_2)^n\Vert ^\{1/n\} \rightarrow 0$ as $n \rightarrow \infty $. Without these conditions, we describe the operators with the same local spectra only in some particular cases.},
author = {Torgašev, Aleksandar},
journal = {Czechoslovak Mathematical Journal},
keywords = {Banach space; spectrum; local spectrum; Banach space; spectrum; local spectrum},
language = {eng},
number = {1},
pages = {77-83},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On operators with the same local spectra},
url = {http://eudml.org/doc/30403},
volume = {48},
year = {1998},
}

TY - JOUR
AU - Torgašev, Aleksandar
TI - On operators with the same local spectra
JO - Czechoslovak Mathematical Journal
PY - 1998
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 48
IS - 1
SP - 77
EP - 83
AB - Let $B(X)$ be the algebra of all bounded linear operators in a complex Banach space $X$. We consider operators $T_1,T_2\in B(X)$ satisfying the relation $\sigma _{T_1}(x) = \sigma _{T_2}(x)$ for any vector $x\in X$, where $\sigma _T(x)$ denotes the local spectrum of $T\in B(X)$ at the point $x\in X$. We say then that $T_1$ and $T_2$ have the same local spectra. We prove that then, under some conditions, $T_1 - T_2$ is a quasinilpotent operator, that is $\Vert (T_1 - T_2)^n\Vert ^{1/n} \rightarrow 0$ as $n \rightarrow \infty $. Without these conditions, we describe the operators with the same local spectra only in some particular cases.
LA - eng
KW - Banach space; spectrum; local spectrum; Banach space; spectrum; local spectrum
UR - http://eudml.org/doc/30403
ER -